If the table was based on half-time scores then Lawrie Sanchez's side would be near the top, but after Simon Davies gave them the lead with a superb volley, the hosts sat back and allowed Kevin Doyle to secure what looked like a deserved point for the visitors.
But American Dempsey popped up at the back post with just under 20 minutes to go, before Healy notched to earn Sanchez some breathing space with only Fulham's second league win of the season.
Fulham started the brighter as Reading disappointed in a poor opening 45 minutes and after a quarter of an hour the hosts took the lead in style.
Davies matched his eye-catching effort against Sunderland last weekend by smashing home a volley from the edge of the area after Hameur Bouazza's cross was only half cleared.
The goal was especially harsh on Reading goalkeeper Marcus Hahnemann as Davies' chance came moments after the American had made a superb reaction save to foil Aaron Hughes.
Ten minutes after half-time, Reading got back on level terms after a period of sustained pressure.
Antti Niemi excelled himself in saving from James Harper and Stephen Hunt before the Finland international showed the other side of his nature when he went missing for a Nicky Shorey corner from the right.
Niemi was a model of indecision as he came, went back and eventually got stuck in no-man's land as Doyle headed in from almost under the crossbar.
But after almost constant Reading pressure, Fulham re-took the lead with Sanchez's two substitutes David Healy and Shefki Kuqi playing crucial roles.
Healy escaped down the right wing and the hulking Kuqi caused panic at the back post, where Dempsey arrived right on cue to poke the ball home, causing Craven Cottage to shake on its riverside foundations.
With Reading chasing an equaliser, Fulham extended their advantage in injury time.
Davies raced down the right and picked out Healy, who gave Hahnemann no chance and left Sanchez jumping for joy on the touchline.
The victory was spoiled somewhat in injury-time, though, as referee Mark Clattenburg, who until then had had a quiet game, harshly sent-off young Fulham left-back Elliot Omozusi for two yellow cards.